The curve between two rows

An estimate that can be wrong by two asked how wrong the count of frozen rotations would have to be for the rotamer account of the gem-dimethyl effect to survive, and answered: wrong by two, which is not an error anybody could make. On the way it dealt with an objection about ester tethers by putting two rows in a table — the ceiling with every rotor free, and the ceiling with one or two of them removed from the count — and said so:

A rotor is treated here as free or absent, and a real hindered rotor is neither.

The two rows are the ends of a curve, and the curve is not a matter of judgement. The ceiling is

$$\left(\frac{1+2x}{2x}\right)^{r}, \qquad x = e^{-g/kT}$$

with $g$ the energy of the conformer a closure needs. A hindered rotor is one whose closable conformer has been stabilised by some amount $h$, so its own $g$ is $g - h$ and its own contribution is the same expression with the same closed form in it. Nothing has to be assumed to fill the table in.

What one rotor is worth #

Two things are worth reading off it, and they point in opposite directions.

The free-or-absent shortcut is better than it claimed, and by an amount worth stating. It removed an ester’s rotor from the count on the argument that the linkage sits in one conformer with its alternative some twenty kilojoules a mole higher. At that energy the rotor contributes 1.000726, so the shortcut is right to a part in fourteen hundred. That is not an approximation whose size was unknown; it is one whose size is now known and is negligible. Twenty kilojoules is about eight times kT at room temperature, and the closable fraction of a hindered rotor falls as the exponential of minus that — a few parts in ten thousand, which is where the part in fourteen hundred comes from.

And the curve is steep exactly where a torsional barrier lives. Half the fall from 3.32 to 1 happens by four kilojoules a mole, which is about a butane gauche interaction. So a rotor that is slightly hindered — by a substituent, by a heteroatom, by anything short of an ester’s conformational lock — is neither free nor absent, and the table had no row for it.

The verdict that is at risk, and the one that is not #

The two verdicts face in opposite directions, and hindrance treats them differently.

The five-membered closure is refuted: its ceiling of 36.46 is below its measured 250-fold acceleration. Hindering any of its rotors lowers the ceiling further — 120.88, 42.09, 17.10, 4.56, 1.42, 1.01 as four rotors are hindered by 0, 1, 2, 4, 8 and 16 kilojoules a mole. So no tether stiffness can rescue it, and the direction of the effect settles that without any argument about how stiff a real tether is.

The six-membered closure is sufficient: its ceiling of 120.88 is above its measured tenfold. That is the verdict at risk, and it is the only tight one.

Where it gives way #

Bisecting for the hindrance that takes the six-membered ceiling below ten:

The second row is the one worth staring at. No amount of hindrance on two of the four rotors can break the verdict — and the margin is nine per cent. Two free rotors are worth 10.99 against a measurement of 10. A gauche energy a tenth larger, or a measurement a tenth larger, and the answer changes.

That is a different kind of robustness from the five-membered case, where the refutation survives an error of two whole rotors. It is also a different kind from the one the tetramethyl measurement enjoys: 11,000 against a ceiling of 36.46 is a margin of a factor of three hundred, and no arrangement of hindrances on any tether can close it. Three verdicts in one argument, with margins of a factor of three hundred, a factor of two rotors, and nine per cent — and the last of them is the one quoted as a success. Both verdicts have been called robust, for different reasons; this essay puts a number on the difference, and the numbers are a factor of two in rotor count against nine per cent in energy.

The table with its middle filled in #

The middle column is not between the outer two in the way a reader would guess. At three rotors hindered by four kilojoules a mole the ceiling is 10.35 — above the measurement, but by three and a half per cent — while the fully-hindered column says 3.32 and the free column says 120.88. A moderate hindrance lands the six-membered closure almost exactly on its own verdict boundary, which neither of the two rows could have shown.

What a hindered rotor is, and what it is not #

The word is doing two jobs in the literature and only one of them is in this model, so it is worth separating them before the numbers are used for anything.

A rotation can be hard to perform — a barrier between conformers, which is what an infrared spectroscopist means by a hindered rotor and what makes a torsion a vibration rather than a free rotation. Or it can be already in the arrangement the ring needs — a preference among conformers, which is what an ester linkage has and what the tether objection is about.

Only the second appears here. The rotamer account is a ratio of equilibrium populations, and an equilibrium population does not know how long anything took: a rotor with a ten kilojoule barrier and three equal minima is a free rotor as far as every number in this essay is concerned. The barrier a force field does compute is a different quantity in a different model.

That distinction is why the ester case is unambiguous. Twenty kilojoules a mole is the energy of the alternative conformer, not a barrier height, and it is exactly the quantity this model wants.

Why hindering lowers the ceiling at all #

It is worth being explicit, because the sign is what makes the five-membered refutation safe and the intuition can run the other way.

The ceiling is a ratio of populations: how much more often a substituted chain is in a closable arrangement than an unsubstituted one. The substituent works by making the extended arrangements expensive, so the ceiling is large when the unsubstituted chain is rarely closable — which is when the closable conformer costs energy.

Hindering the rotor in the sense meant here means making the closable conformer cheaper, so the unsubstituted chain is already curled and the substituent has less to buy. In the limit the rotor sits permanently in the arrangement the ring needs, the substituent buys nothing along that coordinate, and the factor is one.

So a stiffer tether is a tether that has already done the work, which is the opposite of the intuition that a stiff tether is harder to close. That intuition is about a barrier; this is about a preference, and the rotamer account has only preferences in it. The account with the wrong sign failed for a related reason — a mechanism can be real and push the wrong way — and this is the same care taken before the arithmetic rather than after it.

One closed form, used three ways #

Three arguments now rest on the same closed form, and they use it in three different ways, which is worth setting out because it is unusual for one expression to do that much work.

A ceiling that rises where the measurements fall uses it as a prediction: the ceiling rises with ring size while the measured accelerations fall, so the account is refuted at one size and sufficient at another and there is a crossing between them. An estimate that can be wrong by two uses it as a margin: the ceiling is exponential in an integer, so how wrong could the integer be has an answer that is itself an integer. This essay uses it as a sensitivity: the same expression with a continuous parameter in it, and the parameter’s critical value found by bisection.

The three are not interchangeable and the last is the weakest. A prediction can be refuted, a margin in whole rotors cannot be argued away, and a sensitivity in kilojoules a mole is only as good as somebody’s estimate of how hindered a tether is. What this adds is not another verdict but the shape of the surface the existing verdicts sit on — and the surface turns out to be flat under one of them and steep under another.

That is also why the nine per cent matters more than it looks. An argument with two refutations and one sufficiency reads as a pattern; one with two refutations and one near miss reads as a uniform failure with an exception that has not been checked hard enough. Which of those it is depends on a number nobody has measured.

What is quoted, and what is computed #

Two things are quoted and both are measurements. Butane’s gauche energy, 3.8 kJ/mol, and the accelerations themselves: 250-fold for the gem-dimethyl γ-butyrolactone closure, 11,000-fold for the tetramethyl one, tenfold for the δ-valerolactone. The twenty kilojoules a mole for an ester linkage’s alternative conformer is quoted too, and it is used only to show that the shortcut it justifies is safe.

Everything else is computed. The ceiling’s closed form is checked against a direct enumeration over all $3^r$ staggered arrangements at each rotor count — the two must agree to a part in a thousand, and they do — so the interpolation is being done on an expression that has been tested rather than on an expression that was written down.

The critical hindrances are found by bisection on a ceiling that falls monotonically, and the monotonicity is checked rather than assumed: a bisection on a curve that turned around would return a root and say nothing.

What this cannot say #

A single hindrance for every rotor is a simplification. A real tether’s rotations are not equally hindered — the bond next to the ester is not the bond in the middle of the chain — and the sweep here puts the same $h$ on each of a stated number of them. What that buys is a one-parameter family that can be tabulated; what it costs is that the distribution of hindrances across a real tether is not represented at all.

The model has no barrier in it. Everything here is a population, so hindered means a conformer that is high in energy and not a rotation that is slow. A tether whose rotations are genuinely restricted on the timescale of the closure is a different calculation, and it is a kinetic one.

The hindrance is applied to the rotors a closure freezes, and which those are is still an estimate. The whole rotor-count question is that $r = n - 2$ was taken as read, and nothing here computes it either — so a tether whose rotor count is wrong by one has a ceiling wrong by a factor of 3.32 before any hindrance is applied, which is larger than every effect in this essay. The two uncertainties multiply and only one of them has been reduced.

And the ceiling remains a ceiling. It is what the rotamer account gives when the substituent’s penalty is made arbitrarily large, so every number here is an upper bound on that account and not a prediction of any rate. A closure below its ceiling is not explained by the account; it is merely not refuted by it, which is the distinction the ceiling’s first derivation is careful about.

What was checked #

The two ends of the curve are the table’s two rows, to machine precision — no hindrance gives the free-rotor ceiling exactly, and a hindrance of four hundred kilojoules a mole gives the ceiling with those rotors removed from the count exactly. Both are checked against the published numbers rather than against a re-derivation, which is what makes the curve an interpolation of that table rather than of a new one.

The ceiling falls at every step, checked as a monotone sequence, because the bisection depends on it and a bisection cannot detect its own assumption failing.

No hindrance of one rotor can break the six-membered verdict. That is the refusal, and it is the one a careless implementation would get wrong: a function that reported a critical hindrance there would be reporting a root of an equation with no root, which is exactly what a bisection returns when it is not asked whether the answer exists.

And an ester’s contribution is between one and 1.001, checked from both sides. Below one would be arithmetic that has gone through the floor; above 1.001 would mean the shortcut needed defending rather than merely stating.

Which arrangement of a fixed hindrance is worst #

A tether has a list of hindrances rather than one, and the ceiling is a product over the list. So the question that decides whether the curve above is an example or a bound is which arrangement of a fixed total hindrance drives the ceiling lowest — and it is settled rather than guessed, because the shape of one rotor’s contribution is known.

Written out, one rotor contributes

$$f(h) = 1 + c\,e^{-h/kT}, \qquad c = \tfrac{1}{2}e^{g/kT} = 2.316$$

at butane’s 3.8 kJ mol⁻¹, which gives $f(0) = 3.316$ and $f(\infty) = 1$ — the two tabulated rows, recovered as the ends.

The ceiling is $\prod_i f(h_i)$ and its logarithm is $\sum_i \log f(h_i)$. Differentiating twice,

$$\frac{d^2}{dh^2}\log f = \frac{u}{(kT)^2 (1+u)^2} > 0, \qquad u = c\,e^{-h/kT},$$

so $\log f$ is convex in the hindrance, everywhere and for every value of the constants. By Jensen’s inequality a sum of convex terms at fixed total is smallest when the terms are equal. Spreading a given hindrance evenly across the rotors is the worst case for the ceiling, and concentrating it on one rotor is the best.

The sizes are worth seeing, for three rotors at a total hindrance of six kilojoules a mole. Spread evenly, two each, the ceiling is 8.41. Split between two rotors it is 9.48. Loaded entirely onto one it is 13.26. All three sit below the free-rotor 36.46, and the spread of a factor of 1.6 between the extremes is the amount the arrangement is worth.

That converts the sweep above from an illustration into a bound, and the bound runs the useful way. The five-membered refutation needs the ceiling to stay below 250, and every arrangement lowers it — so hindrance of any kind, however distributed, strengthens the refusal rather than threatening it. The six-membered verdict is the one at risk, and its worst case is now identified rather than assumed: an evenly hindered tether, which is also the arrangement a chain of similar bonds is most likely to have.

Still open: a real tether’s hindrances, and the measurement’s own margin #

The distribution the simplification hides is settled in the section above: one rotor’s log-contribution is convex in its hindrance, so an even spread is the worst case and the sweep is a bound rather than an example. What is still an example is the list itself — a real ester tether’s hindrances are three particular numbers rather than a total to be distributed, and measuring them would replace a bound with a value.

The nearer question is the measurement’s own margin. The verdict at nine per cent is the tightest in the argument, and nine per cent is inside the uncertainty of a rate ratio measured in the nineteen-sixties. Recomputing the whole comparison with an explicit uncertainty on the measured tenfold — rather than as a number compared against a number — would say whether the six-membered case is a verdict at all, or whether the honest statement is that the account is not excluded there. The question of what an error of two rotors means has the same shape about its other input, and has never been asked about this one.